Problem 52 - Precalculus with Limits

Question

In Exercises 49 - 58, find the sum using the formulas for the sums of powers of integers. \( \sum_{n=1}^{10}n^3 \)

Step-by-Step Solution

Answer

The sum of the cubes of the first 10 natural numbers is 3025.

Step 1: Understanding and using the formula

The formula for the sum of cubes of the first n natural numbers is \((\frac{n*(n+1)}{2})^2\). Herein, n refers to the number of terms, which in this case is 10.

Step 2: Substituting the value of n into the formula

Now the value n=10 is substituted into the formula \((\frac{n*(n+1)}{2})^2\): \((\frac{10*(10+1)}{2})^2\).

Step 3: Simplifying the formula

The formula simplifies to \((\frac{10*11}{2})^2 = (55)^2\).

Step 4: Final calculation

Finally, the calculation is completed with \(55^2 = 3025\).

Key Concepts

Sum of Cubes Formula

The Sum of Cubes Formula is a fascinating mathematical expression. This formula is used to find the sum of the cubes of the first \( n \) natural numbers. Mathematically, it is expressed as:

• \( S_n = \left(\frac{n(n+1)}{2}\right)^2 \)

Understanding the Formula:

- The expression \( \frac{n(n+1)}{2} \) corresponds to the sum of the first \( n \) natural numbers. - Once you compute this sum, squaring it gives the sum of the cubes. This formula provides a shortcut to calculating cubic sums without adding each individual cube. For example, instead of calculating \( 1^3 + 2^3 + ... + 10^3 \) directly, you use the formula:

• Substitute \( n = 10 \).
• Calculate \( \frac{10(11)}{2} = 55 \).
• Then square it to get \( 55^2 = 3025 \).

This approach is not only efficient but also ensures accuracy, especially when you’re dealing with large numbers.

Natural Numbers

Natural numbers are the foundation of arithmetic and mathematical calculations. These are the numbers that you usually count with, like 1, 2, 3, and so forth. They start from 1 and continue indefinitely. Even though zero is sometimes included, it primarily applies to whole numbers or non-negative integers.

Natural numbers are pivotal when using formulas like the Sum of Cubes Formula. This is because such formulas often require counting from 1 up to a certain point. Understanding the properties of these numbers helps in various computations. A few key properties are:

• They are always positive.
• They have no fractions; every natural number is whole.
• The sequence is infinite.

When working with integer sequences and sums, recognizing and using natural numbers correctly is essential. Examples of natural numbers in practice would include the sum of powers for 1 through 10, as presented in the exercise.

Integer Sequences

Integer Sequences play a crucial role in understanding patterns and formulas in mathematics. Specifically, they are lists of numbers arranged in a particular order, where each number is an integer. These sequences can be finite, like \( 1, 2, 3, \ldots, 10 \), or infinite, like all natural numbers.

Working with Integer Sequences:

- Integer sequences can be uniform, following a simple arithmetic pattern, or more complex. - They are used to simplify larger computations, like finding the sum of cubes.
The reason integer sequences are important is due to their predictability, which allows us to apply formulas such as those for sums of powers. For instance, knowing the sequence from 1 to 10 can easily help evaluate cubic sums using the Sum of Cubes Formula. The sum of powers in sequences illustrates repetitive and calculable patterns, making math more approachable. Recognizing integer sequences is vital in tackling various mathematical problems efficiently.